Here we mainly focus on establishing spectral independence for the Gibbs distributions of interest.
Once this is established, we use results from other works to derive bounds on the mixing time.
Specifically, for the bounded \({\rho }({\bf A}_G)\)  case, which also implies that the maximum
degree \({\Delta }\)  is bounded, we use Theorem 1.9 from [1]} (arxiv version).
For the unbounded cases, see the discussion in Section , in the Appendix.
